WEBVTT mathematics/multivariable-calculus/hovasapian
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Hello and welcome back to educator.com and multi variable calculus.
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Today's lesson we are going to be talking about divergence and curl in 3-space.
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So we have talked about divergence and curl before, we did it for 2-space when we discussed Green's Theorem.
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Now we are going to talk about it in 3-space. We are going to introduce some new symbolism that will take care of all cases all at once.
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Let us just go ahead and jump right on in. Okay.
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So, we will let f(x,y,z) = f1(x,y,z), f2(x,y,z), and f3(x,y,z).
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This is just a highly explicit way of representing a vector field. A vector field, these are the coordinate functions, the coordinate functions are functions of x,y,z themselves. I have just written everything out.
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So, let this be a vector field on an open set s in R3, in 3-space, 3-dimensional space.
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Okay. That is... and I know you know what this is but I am just going to repeat it for the sake of being complete.
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For each point in s, there is a vector given by f pointing in some direction away from that point... pointing in some direction... starting at that point. That is it.
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So, if I take some 3-space, so in this particular case let us just say that our open set s happens to be all of 3-dimensional space.
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If I pick a point at random, well if I pick a point and I put that point in for f, I am going to get a vector going this way, maybe for this point I have a vector this way, maybe for this point a vector this way, that way, that way, that way, that way, could be any number of things. That is a vector field in 3-space. That is all it is.
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Okay, let us define the divergence of f.
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The divergence of f, which is symbolized as div(f) = df1 dx + df2 dy + df3 dz, or in terms of capital D notation, D1f1 + D2f2, + D3f3. Okay.
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It is a scalar, it is a number. It is a scalar. A number at a particular point, Df dx + Df2 dy + Df3 dz at a particular point you put that particular point into this thing and it is just going to spit out a number.
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Okay. That is the divergence in 3-space. It is just the analog of the divergence in 2-space.
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So the curl is the... it is the analog of the curl in 2-space that we discussed previously, but it is a little bit more notationally complicated.
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So, let us go ahead and write it out, and then we will go ahead and give you a symbolic way of representing it.
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So, curl of f... define the other definition, the curl of the vector field... is equal to... this one I am going to do the capital D notation first, and then I will do the regular partial derivative notation.
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It is going to be D2f3 - D3f2, D3f1 - D1f3, D1f2 - D2f1. Notice this is not a scalar. This is a vector. The curl of a vector field gives you another vector field.
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In terms of partial derivative notation, this looks like this. So, D2 of f3, this is Df3 dy - Df2 dz.
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This is Df1 dz - Df3 dx.
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This is Df2 dx - Df1 dy.
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I hope to heaven I have got all of those correct. Okay. That is the curl. It is a vector.
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So, when you are given a vector field f, when you take the divergence of it, you want to put a number at a given point.
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When you take the vector field f, the curl of that vector field, what you end up with is another vector field. It is a vector at a given point.
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Okay. Now we are going to introduce something called the Del operator.
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The del operator is a symbolic way of simplifying the calculations. That is really what it is. And making things just look more elegant. The del operator, okay, it is usually an upside down triangle, or you can just write del.
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So, the definition of the operator is the following. For the time being, the notion of an operator, an operator is just something that tells you to do something to something else.
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In other words, we talk about the differential operator d dx. So, for example you knew what this is, d dx. If I apply this differential operator to some function f, d dx of f, well, that is just df dx.
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That just means take the derivative of it. The integral operator. The integral operator says integrate on the function f, you are going to end up getting something else, so that is all an operator is.
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It is a fancy term that says do something to this function. Well, the del operator is the same thing.
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It is an operator that says do this to a given function, except it has multiple parts. These are individual operators. The differential, the integral operators. The del operator actually is written symbolically in the form of a vector.
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So, but other than that you treat it exactly the same way as you do anything else. It says do something to something.
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As it turns out, the del operator is a differential operator. It is a partial differential operator. You will see what we mean in just a minute.
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So, let us go ahead and write out the symbol and then do some examples and of course everything will make sense.
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Let us see, so this, or del, is equivalent to d dx, d dy, d dz, or D1, D2, D3, notice there is no function here because it is an operator.
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I have to choose a function and then say do this to it. That is the whole idea. We write it as a vector because it is going to operate as a vector on another vector. That is the whole idea.
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So, now, let us go ahead and write what we mean by divergence and curl symbolically, using this del operator notation.
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Okay. So, symbolically, and again, this is all symbolic. Symbolically, the divergence of f = the del operator dotted with the vector field, f.
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Well, the del operator dotted with the vector field f, is equal to... well, the del operator is D1, D2, D3, that is my symbolic operator for del, dotted with well... f is f1, f2, f3.
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Well, I know what a dot product is. It is just this × this, this × this, this × this, added together.
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Except I am multiplying these two numbers, this is a symbolic vector operation. It says do d1 to f1, in other words, take the derivative of f1 with respect to x. Take the derivative of f2 with respect to y, take the derivative of f3 with respect to z. This is a symbolic notion.
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We are symbolizing using the idea of a vector, that is what an operator is. So, this is equal to... so, it is d1f1 + d2f2 + d3f3, except this is not multiplication, this d1f1... this is a differential operator.
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It says take the derivative of f1 with respect to the first variable, this says take the derivative of f2 with respect to the second variable, this says take the derivative of f3 with respect to the 3rd variable. I hope this makes sense.
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Now the curl of f. Okay. This is going to be kind of interesting. Let me go to the next page.
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So, the curl of f. Now you can memorize this any way you want to. If you want to just go back to what I initially wrote on the first page of this lesson, when I defined divergence and curl, you are more than welcome to remember curl in that way if you want to remember the indices -- 23 32, 13 31, 12 21.
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That is fine, or you can remember it this way, symbolically.
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So, the curl of f, it is defined as del cross f, the del operator crossed with the vector field. Well we know what a cross product is. We have been dealing with it symbolically. It is the symbolic determinant i,j,k, and now del cross f... it is like this is a vector, this is a vector.
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Well, the first vector is in the second row, so we will just write D1 D2 D3, and f is just f1 f2 f3.
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We will symbolically take the determinant of that. When you do that, you end up with the following.
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You end up with (D2f3 - D3f2)i - (D1f3 - D3f1)j + (D1f2 - D2f1)k. i... j... k... this is the first component function, second component function, third component function.
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What you end up with is exactly what we had earlier. You get D2f3 - D3f2, this - turns this into a negative, turns this into a positive... what you end up with is D3f1 - D1f3 and you get D1f2 - D2f1.
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This is the vector representation, this is the i,j,k, representation. This is the curl.
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So, all I have done is I have taken this... I have created this thing called the del operator, given it a symbol like a vector, and I have been able to define the divergence and curl in terms of the two vector operations that I have.
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The divergence of the vector field f is equal to del ⋅ f, and then curl of the vector field f is equal to del cross f.
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This is the symbolic way of keeping things straight. So, let us just do some examples and I think it will make sense here. So, Example 1.
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We will let our vector field f = the first component function is x², the second component function is xy, and our third component function is going to be e^x,y,z.
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Okay. So, the divergence of f, divergence is always going to be easier because you are just sort of taking partial derivatives 1 by 1. So, the divergence, the partial of this with respect to x is going to be 2x, + the partial of this with respect to y, so this is going to be cos(xy) × x, so it is going to be + x × cos(xy), and the partial of this with respect to z is just going to be e^cy.
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There you go. That is the divergence. It is a scalar, not a vector.
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Now, let us go ahead and form the curl of f. The curl of f, well, we said that the curl of f = del cross f, so let us go ahead and form our symbolic determinant here... i, j, k.
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We have d1, d2, d3, or if you want, let us go ahead this time for this particular example, let us use our partial differential and our d dx, d dy, it does not really matter.
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So, let us do i, j, k, so we have d dx, d dy, an d dz, that is the del vector operator, and then we have f, which is x², sin(xy), and e^xyz. We want to form this determinant.
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Okay. So, let us see what we get. When we expand along the first row, we get the derivative of this with respect to y - the derivative of this with respect to z, so I end up getting the derivative of this with respect to y is e^xz, e^x × z - 0 × i - ... because remember it is + - +, alternating sign.
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The derivative of this with respect to x - the derivative of this with respect to z.
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So the derivative of this with respect to x is e^xyz is e² × yz - 0, and that is going to be j.
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Of course our last one k is going to be the derivative of this with respect to x - the derivative of this with respect to y.
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So, the derivative of this with respect to x is going to be y × cos(xy) - 0 × k, so is the... so I get this, this, and this... and so I end up with e^x × z is my first component function of my curl, e^x × yz is the second component function of my curl, and y × cos(xy) is the third component of my curl vector. There we go.
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This is a scalar divergence... curl is a vector at a given point.
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If I take all of the points, it gives me a vector field. That is it.
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Okay. Let us do another example. So, example 2.
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f(x,y,z) = x²yz, xy³z, and xyz⁴. Okay, well, let us see what we have got.
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The divergence of f, we said, is equal to del ⋅ f, well del ⋅ f is d1f1 + d2f2 + d3f3, but we are not multiplying the d and the f, this means take the derivative of f.
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Well, the derivative of this with respect to x is going to be 2xyz + this one, the derivative with respect to y is going to be + 3xy²z, and the derivative with respect to z of this one is going to be +4xyz³. This is our divergence of the particular vector field.
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Okay. So, now let us go ahead and do the curl of the vector field. The curl of this vector field, well it is equal to del cross f, and del cross f, well, is symbolic.
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It is going to be i, j, k... let us do capital D notation here... D1, D2, D3, and we have x²yz, we have xy³z, and we have xyz⁴.
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Okay. So, now let us go ahead and expand along the first row. It is going to be the derivative with respect to y of this - the derivative with respect to z of this. The derivative with respect to y of this is xz⁴.
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The derivative with respect to z of this is xy³. This is the i component... - , now we go to the next one.
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The derivative with respect to x of this - the derivative with respect to z of this, derivative with respect to the first variable which is x, derivative with respect to the third variable which is z. That is what is going on here.
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This × this, this × this, except it is not times, it is symbolic. It means operate on this.
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Okay. So, the derivative with respect to x of this is yz⁴ - the derivative with respect to z of this, x²y. This is the j component.
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Okay, we are almost done. Now, the derivative with respect to x of this - the derivative with respect to y of this.
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So, the derivative with respect to x of this is going to be y³z - the derivative with respect to y of this is going to be x²z.
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Again, I hope that you are confirming this for me. There are lots of x's, y's, z's, i's, j's, k's, floating around. y³z, there you go.
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I will go ahead and actually leave it in this form. That is the first coordinate function of the curl, that is the second coordinate function of the curl, and that is the third coordinate function of the curl.
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Actually, you know what, let me go ahead and write it out. It is not a problem.
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So, I will go ahead and erase these stray lines here.
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So, we have got xz⁴ - xy³, that is the first component function, we have yz⁴ -x²y, and then we have y³z - x²z, notice the divergence is a scalar, the curl is a vector.
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It has three component functions... an x, a y and a z. At a given point (x,y,z), there is some vector pointing in some direction and away from that point. That is the whole idea.
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Again, it is all based on this notion of what we call an operator. It is just a symbolic way of telling you what to do to a given function.
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It is a unifying scheme, so you had this thing. You can take the divergence and curl of a vector field. We want to be able to express that in terms that we know.
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Well, if we gave this... we called it a del operator, we have it a symbolic representation s D1, D2, D3, as a symbolic vector... If we do del ⋅ f, we get divergence of f, if we get del cross f, we get the curl of f. That is it. It is just a unifying scheme.
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Okay, thank you for joining us here at educator.com for divergence and curl. We will see for a discussion of the divergence theorem in 3-space. Take care, bye-bye.